|Description: Axiom of Specialization.
A quantified wff implies the wff without a
quantifier (i.e. an instance, or special case, of the generalized wff).
In other words if something is true for all x, it is true for any
specific x (that would
typically occur as a free variable in the wff
substituted for φ).
(A free variable is one that does not occur in
the scope of a quantifier: x and y are both free in x = y,
but only x is free in
∀yx = y.) Axiom scheme C5' in [Megill]
p. 448 (p. 16 of the preprint). Also appears as Axiom B5 of [Tarski]
p. 67 (under his system S2, defined in the last paragraph on p. 77).
Note that the converse of this axiom does not hold in general, but a
weaker inference form of the converse holds and is expressed as rule
ax-gen 1270. Conditional forms of the converse are given
by ax-12 1336,
ax-16 1605, and ax-17 1352.
Unlike the more general textbook Axiom of Specialization, we cannot choose
a variable different from x for the special case. For use, that
requires the assistance of equality axioms, and we deal with it later
after we introduce the definition of proper substitution - see stdpc4 1569.
(Contributed by NM, 5-Aug-1993.)